| L(s) = 1 | − 0.779·2-s − 2.96·3-s − 1.39·4-s + 2.30·6-s − 0.937·7-s + 2.64·8-s + 5.77·9-s + 1.14·11-s + 4.12·12-s + 5.29·13-s + 0.730·14-s + 0.724·16-s + 4.50·17-s − 4.49·18-s − 5.43·19-s + 2.77·21-s − 0.895·22-s − 1.97·23-s − 7.83·24-s − 4.12·26-s − 8.21·27-s + 1.30·28-s − 7.61·29-s − 9.12·31-s − 5.85·32-s − 3.40·33-s − 3.51·34-s + ⋯ |
| L(s) = 1 | − 0.551·2-s − 1.71·3-s − 0.696·4-s + 0.942·6-s − 0.354·7-s + 0.934·8-s + 1.92·9-s + 0.346·11-s + 1.19·12-s + 1.46·13-s + 0.195·14-s + 0.181·16-s + 1.09·17-s − 1.06·18-s − 1.24·19-s + 0.605·21-s − 0.190·22-s − 0.412·23-s − 1.59·24-s − 0.809·26-s − 1.58·27-s + 0.246·28-s − 1.41·29-s − 1.63·31-s − 1.03·32-s − 0.592·33-s − 0.602·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 0.779T + 2T^{2} \) |
| 3 | \( 1 + 2.96T + 3T^{2} \) |
| 7 | \( 1 + 0.937T + 7T^{2} \) |
| 11 | \( 1 - 1.14T + 11T^{2} \) |
| 13 | \( 1 - 5.29T + 13T^{2} \) |
| 17 | \( 1 - 4.50T + 17T^{2} \) |
| 19 | \( 1 + 5.43T + 19T^{2} \) |
| 23 | \( 1 + 1.97T + 23T^{2} \) |
| 29 | \( 1 + 7.61T + 29T^{2} \) |
| 31 | \( 1 + 9.12T + 31T^{2} \) |
| 37 | \( 1 - 5.51T + 37T^{2} \) |
| 41 | \( 1 - 6.55T + 41T^{2} \) |
| 43 | \( 1 - 0.588T + 43T^{2} \) |
| 47 | \( 1 + 2.18T + 47T^{2} \) |
| 53 | \( 1 + 1.78T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 - 13.0T + 67T^{2} \) |
| 71 | \( 1 + 3.64T + 71T^{2} \) |
| 73 | \( 1 - 7.30T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 + 3.52T + 89T^{2} \) |
| 97 | \( 1 + 6.10T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.71467819338197560567364000475, −6.93824233190982570775527028751, −6.08090244550342669840003288336, −5.77151301919342209702341296274, −4.99084232571257336519551547435, −4.03843004781108251535626772009, −3.67555452086978177028955097610, −1.77493822026476317732142218906, −0.951052393975359793933545907184, 0,
0.951052393975359793933545907184, 1.77493822026476317732142218906, 3.67555452086978177028955097610, 4.03843004781108251535626772009, 4.99084232571257336519551547435, 5.77151301919342209702341296274, 6.08090244550342669840003288336, 6.93824233190982570775527028751, 7.71467819338197560567364000475
